Transitions in the Unitary Matrix Models

Abstract

The first-, second- and third-order phase transitions, or discontinuities, in the unitary matrix models will be discussed in this chapter. The Gross-Witten third-order phase transition is described in association with the string equation in the unitary matrix model, and it will be generalized by considering the higher degree potentials. The critical phenomena (second-order divergences) and third-order divergences are discussed similarly to the critical phenomenon in the planar diagram model, but a different Toda lattice and string equation will be applied here by using the double scaling method. The discontinuous property in the first-order transition model of the Hermitian matrix model discussed before will recur in the first-order transition model of the unitary matrix model, indicating a common mathematical background behind the first-order discontinuities. The purpose of this chapter is to further confirm that the string equation method can be widely applied to study phase transition problems in matrix models, and that the expansion method based on the string equations can work efficiently to find the power-law divergences considered in the transition problems.